Instructions Read First: The Following Worksheets

Instructions Read Firstinstructions The Following Worksheets Descri

The following worksheets describe two problems – the first problem is for independent samples and the second problem is for dependent samples. Your job is to demonstrate the solution to each scenario by showing how to work through each problem in detail. You are expected to explain all of the steps in your own words.

Independent Samples Problem:

A researcher conducted a test to learn the effect of lead levels in human bodies. He collected the IQ scores for a random sample of subjects with low lead levels in their blood and another random sample of subjects with high lead levels in their blood. The summary of findings is listed below. Use a 0.05 significance level to test the claim that the mean IQ score of people with low lead levels is higher than the mean IQ score of people with high lead levels. We do not know the values of the population standard deviations.

• Low Lead Level: n1 = 60, s1 = 17.45
• High Lead Level: n2 = 7, s2 = 9.68

Dependent Samples Problem 1:

Days of Release/Book comparison between Phoenix and Prince:

• Phoenix: 44.3, 22.8, 29.0, 10.3, 7.6
• Prince: 58.4, 28.3, 21.9, 9.4, 6.2

Use a 0.05 significance level to test his claim that the Prince movie made more money at the box office than the Phoenix movie, based on the first few days' grossed amounts in millions.

Dependent Samples Data and Questions:

The Harry Potter books and movies made a lot of money. A fan wanted to learn which of his favorite movies made more money. He collected the amounts grossed in millions during the first few days of releases of the movies Harry Potter and the Half-Blood Prince and Harry Potter and the Order of the Phoenix. Use a 0.05 significance level to test his claim that the Prince movie did better at the box office. Use the p-value method to determine whether or not to reject the null hypothesis and state your conclusion.

Paper For Above instruction

The provided worksheets present two statistical scenarios involving hypothesis testing. The first involves independent samples assessing whether low lead levels correlate with higher IQ scores compared to high lead levels. The second involves dependent samples comparing box office earnings of two Harry Potter movies over initial days. Each scenario requires formulating hypotheses, conducting appropriate statistical tests, calculating critical values, test statistics, and p-values, and then interpreting the results to make informed conclusions.

Analysis of Independent Samples: Lead Levels and IQ Scores

The first scenario examines whether children with lower lead exposure tend to have higher IQ scores than those with higher lead exposure. The hypotheses for this test are set up as:

• Null hypothesis (H₀): μ₁ ≤ μ₂, stating that the mean IQ of the low lead group is less than or equal to that of the high lead group.
• Alternative hypothesis (H₁): μ₁ > μ₂, suggesting that the low lead group has a higher mean IQ.

Since we do not know the population standard deviations, this is a two-sample t-test for the difference of means. The test is right-tailed because the claim specifically posits that the mean IQ score of the low lead group exceeds that of the high lead group.

Calculating the critical value involves identifying the degrees of freedom using the Welch-Satterthwaite equation, due to unequal variances. With a significance level (α) of 0.05, the critical t-value is determined from t-distribution tables or software.

The test statistic is calculated as:

t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂)

where x̄₁ and x̄₂ are the sample means, s₁ and s₂ are the sample standard deviations, and n₁ and n₂ are the sample sizes. Plugging in the known values results in a t-value that is then compared against the critical value. The p-value, representing the probability of observing such a statistic under the null hypothesis, is calculated using statistical software or t-distribution tables.

Based on the computed values, if the test statistic exceeds the critical value and the p-value is less than 0.05, the null hypothesis is rejected, indicating evidence that lower lead levels are associated with higher IQ scores.

Analysis of Dependent Samples: Box Office Revenue Comparison

The second scenario compares the first few days' gross earnings of the Harry Potter and the Half-Blood Prince and Harry Potter and the Order of the Phoenix movies. The hypotheses are:

• H₀: μ_d = 0 (no difference in mean earnings)
• H₁: μ_d > 0 (Prince movie earns more on average)

This is a paired t-test, suitable because the data are dependent (the same periods are compared for two movies). The analysis begins by calculating the differences in earnings for each paired observation, then computing the mean and standard deviation of these differences.

The test statistic is:

t = (mean difference) / (standard deviation of differences / √n)

where n is the number of paired observations. The degrees of freedom are n - 1. The critical t-value for α = 0.05 is obtained from statistical tables, and the p-value is calculated accordingly. If the test statistic exceeds the critical value and the p-value is less than 0.05, the null hypothesis is rejected, supporting the claim that the Prince movie grossed more initially.

Analysis of Correlation and Regression: Harry Potter Box Office Data

Additional data involving the relation between movie gross earnings and other variables requires calculating the Pearson correlation coefficient (r), which measures the strength and direction of the linear relationship. To find r, the covariance of the variables is divided by the product of their standard deviations.

Once r is calculated, it is compared to the critical value for correlation at α = 0.05, determined based on sample size. A significant r indicates a meaningful linear association.

Regression analysis involves fitting a line to predict earnings based on variables like days or release order, using least squares estimation. The slope indicates the average change in earnings per unit change in the predictor, while the y-intercept is the expected earnings when the predictor is zero.

The regression equation allows predicting the movie earnings for given values, such as estimating the expected gross for a movie with a given attribute. Its appropriateness as a model depends on the strength of the correlation and the residual analysis. A strong correlation and random residuals suggest a good fit.

Conclusions

Through hypothesis tests and regression analysis, the worksheets demonstrate the application of statistical methods to real-world scenarios. In the lead-IQ study, sufficient evidence may support the claim that lower lead exposure correlates with higher intelligence. The box office comparison can verify whether one movie indeed outperformed the other initially. Regression analysis helps understand relationships and make predictions, providing insights useful in decision-making processes across fields.

References

• Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.
• Gravetter, F. J., & Wallnau, L. B. (2016). Statistics for the Behavioral Sciences. Cengage Learning.
• Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. W. H. Freeman.
• Siegel, S., & Castellan, N. J. (1988). Nonparametric Statistics for the Behavioral Sciences. McGraw-Hill.
• McClave, J. T., & Sincich, T. (2018). A First Course in Business Statistics. Pearson.
• Ross, S. M. (2014). Introductory Statistics. Academic Press.
• Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
• Agresti, A., & Finlay, B. (2009). Statistical Methods for the Social Sciences. Pearson.
• Ott, R. L., & Longnecker, M. (2015). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.
• Yates, F., Moore, D. S., & McCabe, G. P. (2017). The Practice of Statistics. W. H. Freeman.